⭕ Circles & Power of a Point

Circles are fundamental in AMC geometry, and the Power of a Point theorem unifies many circle relationships into one powerful tool.

🎯 Key Concepts

Basic Circle Properties

  • Chord: Line segment connecting two points on the circle
  • Tangent: Line touching the circle at exactly one point
  • Secant: Line intersecting the circle at two points
  • Radius: Distance from center to any point on the circle
  • Diameter: Longest chord, passing through the center

Power of a Point Theorem

Fundamental: For any point $P$ and circle, the product of distances from $P$ to two intersection points of any line through $P$ with the circle is constant.

Three Cases:

  1. Point inside circle: $PA \cdot PB = PC \cdot PD$ (chords)
  2. Point outside circle: $PA \cdot PB = PC \cdot PD$ (secants)
  3. Point on circle: $PA \cdot PB = PC \cdot PD$ (tangent-secant)

Special Cases

  • Tangent-Secant: $PA^2 = PB \cdot PC$ (when one intersection is tangent)
  • Two Tangents: $PA = PB$ (equal tangent lengths from external point)

🔍 Micro-Examples

Chord Length

In circle with radius 5, if chord is 8 units from center, then chord length is $2\sqrt{5^2 - 4^2} = 2\sqrt{9} = 6$.

Tangent-Secant

If tangent from external point is 6 and secant segment is 4, then other secant segment is $\frac{6^2}{4} = 9$.

Two Tangents

From external point, two tangents to circle are equal in length.

⚠️ Common Traps

Pitfall: Confusing chord and secant

  • Fix: Chord is segment, secant is line

Pitfall: Wrong Power of a Point setup

  • Fix: Always use the correct case (inside, outside, or on circle)

Pitfall: Forgetting tangent properties

  • Fix: Tangent is perpendicular to radius at point of contact

Pitfall: Wrong chord length formula

  • Fix: Chord length = $2\sqrt{r^2 - d^2}$ where $r$ is radius, $d$ is distance from center

🎯 AMC-Style Worked Example

Problem: In the figure, $PA$ is tangent to the circle at $A$, $PB$ is a secant intersecting the circle at $B$ and $C$, and $PA = 6$, $PB = 4$. Find $BC$.

Solution: Using the Power of a Point theorem for the tangent-secant case:

$PA^2 = PB \cdot PC$

Substituting the given values: $6^2 = 4 \cdot PC$ $36 = 4 \cdot PC$ $PC = 9$

Since $PC = PB + BC$ and $PB = 4$: $9 = 4 + BC$ $BC = 5$

Answer: $BC = 5$

🔗 Related Topics

💡 Quick Reference

Circle Formulas

  • Circumference: $C = 2\pi r$
  • Area: $A = \pi r^2$
  • Chord length: $2\sqrt{r^2 - d^2}$ (where $d$ is distance from center)
  • Arc length: $s = r\theta$ (where $\theta$ is in radians)

Power of a Point Cases

  • Inside: $PA \cdot PB = PC \cdot PD$ (chords)
  • Outside: $PA \cdot PB = PC \cdot PD$ (secants)
  • Tangent-Secant: $PA^2 = PB \cdot PC$

Special Properties

  • Tangent: Perpendicular to radius at point of contact
  • Two Tangents: Equal lengths from external point
  • Diameter: Longest chord, subtends right angle

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